An Introduction to Artificial Intelligence

Lecture 4a: Informed Search and ExplorationRamin Halavati (halavati@ce.sharif.edu)

In which we see how information about the state space can prevent algorithms from blundering about the dark.

Outline• Best-first search• Greedy best-first search• A* search• Heuristics• Local search algorithms• Hill-climbing search• Simulated annealing search• Local beam search• Genetic algorithms

UNINFORMED?• Uninformed:

– To search the states graph/tree using Path Path CostCost and Goal TestGoal Test.

INFORMED?• Informed:

– More data about states such as distance to goal.

– Best First Search• Almost Best First Search• Heuristic

– h(n): estimated cost of the cheapest path from n to goal. h(goal) = 0.

– Not necessarily guaranteed, but seems fine.

Greedy Best First Search• Compute estimated distances to goal.

• Expand the node which gains the least estimate.

Greedy Best First Search Example• Heuristic: Straight Line Distance (HSLD)

Greedy Best First Search Example

Properties of Greedy Best First Search• Complete?

– No, can get stuck in loop.

• Time? – O(bm), but a good heuristic can give

dramatic improvement

• Space? – O(bm), keeps all nodes in memory

• Optimal? – No, it depends

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A* search• Idea: avoid expanding paths that are

already expensive

• Evaluation function f(n) = g(n) + h(n)– g(n) = cost so far to reach n– h(n) = estimated cost from n to goal– f(n) = estimated total cost of path through

n to goal–

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A* search example

A* vs Greedy

Admissible Heuristics• h(n) is admissible:

– if for every node n,

h(n) ≤ h*(n),

h*(n): the true cost from n to goal.

– Never Overestimates.– It’s Optimistic.

– Example: hSLD(n) (never overestimates the actual road distance)

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A* is Optimal• Theorem: If h(n) is admissible, A* using TREE-SEARCH is optimal

–TREE-SEARCH: To re-compute the cost of each node, each time you reach it.

–GRAPH-SEARCH: To store the costs of all nodes, the first time you reach em.

Optimality of A* ( proof )• Suppose some suboptimal goal G2 has been generated and is

in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G.

• f(G2) = g(G2) since h(G2) = 0 • g(G2) > g(G) since G2 is suboptimal • f(G) = g(G) since h(G) = 0 • f(G2) > f(G) from above • h(n) ≤ h* (n) since h is admissible• g(n) + h(n) ≤ g(n) + h*(n) • f(n) ≤ f(G)

Hence f(G2) > f(n), and A* will never select G2 for expansion

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Consistent Heuristics• h(n) is consistent if:

– for every node n, – every successor n' of n generated by any action a, – h(n) ≤ c(n,a,n') + h(n')

• Consistency:– Monotonicity– Triangular Inequality.– Usually at no extra cost!

• Theorem: If h(n) is consistent, A* using GRAPH-SEARCH is optimal––

Optimality of A*

• A* expands nodes in order of increasing f value

• Gradually adds "f-contours" of nodes

• Contour i has all nodes with f=fi, where fi < fi+1

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Properties of A*• Complete? Yes (unless there are infinitely

many nodes with f ≤ f(G) )

• Time? Exponential

• Space? Keeps all nodes in memory (bd)

• Optimal? Yes

• A* prunes all nodes with f(n)>f(Goal).• A* is Optimally Efficient.

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How to Design Heuristics?

E.g., for the 8-puzzle:• h1(n) = number of misplaced tiles• h2(n) = total Manhattan distance

(i.e., no. of squares from desired location of each tile)

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Admissible heuristics

• h1(n) = Number of misplaced tiles• h2(n) = Total Manhattan distance

Effective Branching Factor• If A* finds the answer

– by expanding N nodes, – using heuristic h(n), – At depth d,

– b* is effective branching factor if:• 1+b*+(b*)2+…+(b*)d = N+1

Dominance• If h2(n) ≥ h1(n) for all n (both admissible)

• then h2 dominates h1.

• h2 is better for search.

• h2 is more realistic.

• h (n)=max(h1(n), h2(n),… ,hm(n))

• Heuristic must be efficient.

How to Generate Heuristics?• Formal Methods

– Relaxed Problems– Pattern Data Bases

• Disjoint Pattern Sets

– Learning

• ABSOLVER, 1993– A new, better heuristic for 8 puzzle.– First heuristic for Rubik’s cube.

“Relaxed Problem” Heuristic• A problem with fewer restrictions on the actions.

• The cost of an optimal solution to a relaxed problem is an admissible heuristic for the original problem.

• 8-puzzle:– Main Rule:

• A tile can be moved from square A to B if A is horizontally or vertically adjacent to B and B is empty.

– Relaxed Rules:• A tile can move from square A to square B if A is adjacent to

B. (h2)• A tile can move from square A to square B if B is blank. • A tile can move from square A to square B. (h1)

“Sub Problem” Heuristic• The cost to solve a subproblem.

• It IS admissible.

“Pattern Database” Heuristics• To store the exact solution cost to

some sub-problems.

“Disjoint Pattern” Databases• Disjoint Pattern Databases.

– To add the result of several Pattern-Database heuristics.

• Speed Up: 103 times for 15-Puzzle and 106 times for 24-Puzzle.

• Separablity: Rubik’s cube vs. 8-Puzzle.

Learning Heuristics from Experience• Machine Learning Techniques.

• Feature Selection– Linear Combinations

BACK TO MAIN SEARCH METHOD • What’s wrong with A*? It’s both Optimal

and Optimally Efficient.

– MEMORY

Memory Bounded Heuristic Search• Iterative Deepening A* (IDA*)

– Similar to Iterative Deepening Depth First Search

– Bounded by f-cost.

– Memory: b*d

Recursive Best First Search• Main Idea:

– To search a level with limited f-cost, based on other open nodes with continuous update.

Recursive Best First Search

Recursive Best First Search, Sample

Recursive Best First Search, Sample• Complete? Yes, given enough space.

• Space? b * d

• Optimal? Yes, if admissible.

• Time? Hard to analyze. It depends…

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Memory, more memory…• A*: bd

• IDA*, RBFS: b*d

• What about exactly 10 MB?

Memory-Bounded A*• MA*

• Simplified Memory Bounded A* (SMA*)– To store as many nodes as possible (the

A* trend).– When memory is full, remove the worst

current node and update its parent.

SMA* Example

SMA* Code

To be continued…